We cannot know for sure whether religion and family life strengthened or weakened slavery. We can only speculate. There are ways to argue that these things both strengthened and weakened slavery and there is no way to objectively determine which effect was stronger.
Family and religion both helped slaves to feel more of a sense of community with one another and more of a sense that they were regular human beings. Slavery was, of course, a horrible thing. However, family and religion helped to make it somewhat less horrible. They gave slaves a sense of belonging and they allowed slaves to feel like there was a sphere in which they were regular human beings, not just slaves who were seen as property.
It is possible to argue that this strengthened slavery as an institution. If slaves had not had family or religion, they would presumably have been less happy. They might have felt that they had no reason to live or no reason to obey their owners. They might have felt desperate enough to try more escapes or more slave rebellions. Thus, by keeping slaves more contented, family and religion could have strengthened slavery.
On the other hand, we can argue that family and religion weakened slavery. Religion gave slaves a way to resist what the owners told them. It gave them a way to organize themselves and to possibly help one another escape. We can argue that slaveowners would have preferred not to have any influences like those of religion or family on their slaves.
I would argue that, on the whole, slavery was strengthened by family and religion, but there is no way to objectively measure whether this is true.